Effect of Die Temperature on the Flow of Polymer Melts Part I: Flow Inside the Die

BING YANG and L. JAMES LEE'

Depar tment of Chemical Engineering T h e Ohio S t a t e University

Columbus , Ohio 4321 0

The effect of die wall temperature on the flow of polymer melts in circular capillary dies was studied. At constant flow rates, it was found that die wall temperature had a greater effect on the pressure drop than melt temperature. A capillary die with two circular channels with different diameters was designed to simulate the profile extrusion. Changes of wall temperature varied the flow rate ratio between the two channels. An implicit finite difference method was used to simulate the velocity and temperature profiles inside the die. Values predicted by this model matched well with experimental data for both dies.

INTRODUCTION eat transfer to polymer melt flow is very H important in many industrial applications.

For example, control of local extrudate dimen- sions through the adjustment of die flow and extrudate swell by manipulating the local die temperature and melt temperature has been used routinely in the film extrusion (1, 2) and profile extrusion (3, 4). If the film thickness at any local point varies from the set point, one can manipulate the local die temperature to adjust the thickness of the film. For profile extrusions, one can adjust melt temperature and die temperature to vary the shape of extru- date (3). These adjustments are gained from experience and have been used for a long time in the polymer industry. There is, however, a lack of thorough understanding of the mecha- nism involved in these adjustments.

Understanding the polymeric flow through a die requires analysis of flow in a confined chan- nel, which can be divided into three distinct flow regions: entrance flow, fully developed flow, and exit flow.

Heat transfer to the flow of laminar, non- Newtonian polymer melt inside a confined channel is controlled by contributions from sev- eral sources. In addition to the heat conduction and convection mechanisms, temperature sen- sitive transport properties (especially viscosity), the shear rate dependence of viscosity, and vis- cous heating need to be considered (5-10). I t is difficult to isolate the contribution from each source. Most workers have attempted to solve this problem either analytically by assuming

To whom the correspondence should be addressed

temperature independent rheological properties (1 1 - 17) or by numerical solutions to the coupled equations of continuity, momentum, and en- ergy with temperature dependent viscosity (1 8- 28). Experimental measurements of non-iso- thermal laminar flow of polymer melts or solu- tions have also been carried out by many re- searchers (9, 16, 17, 19-21, 23, 24, 29-31). Most previous works, however, only dealt with flow channels with simple geometry and con- sidered mainly the correlation of heat transfer coefficient from experimental data.

In the profile extrusion, a die can be consid- ered as a combination of several simple flow channels. When die temperature changes, the relative flow rates may vary due to the change of temperature-dependent rheological proper- ties. This work analyzes heat transfer to the flow of polymer melt inside dies with a circular channel and two circular channels with differ- ent diameters. The result is to elucidate the effect of die temperature on the extrusion of profiles.

EXPERIMENTAL

Materials and Rheological Characterization A general-purpose, commercial polystyrene

was used in this study. The non-Newtonian shear viscosity and the first normal stress dif- ference, N 1 , for the polymer melt were deter- mined at three temperatures: 210", 220", and 230°C at low shear rates with a Weissenberg Rheogoniometer (Model R18). The non-Newton- ian shear viscosity was also measured at higher shear rates using an Instron capillary rheome- ter. Figure 1 plots the measured shear viscosity

POLYMER ENGINEERING AND SCIENCE, MID-AUGUST, 1987, Yo/. 27, No. 14 1079

B. Yang and L. James Lee

01 azo*cl/ - , .I..... I . ....."I ?".,, ~ ,..,..,, # ,,.....-I 0

10-l l o o 10' l o 2 l o s 10' SHEAR RATE ( l /sec)

Fig. 1. Shear viscosity and f i rs t normal stress dflerence versus shear rate of polystyrene at 210, 220, and 230°C.

EXPERIMENTAL SETUP

PLUNGER

and N1 versus shear rate at three different tem- peratures.

In this study, the viscosity of polystyrene used can be expressed by a Carreau type of equation (32)

(1) 1.1 [ l -n] / l . l 7 = TO,T[ l + ( h T O . T j / / T ) 1

where

~ o , T = 3.6 X 103exp((AT + B)(T - 483.15))

(Pa-s), with

A = 0.66645 X loT3 (K-')

B = -0.38543 (K-I)

X = 0.19998 X lov2 (K.Pa-')

n = 0.30

j / is shear rate (s-') and T is temperature (K) This model can fit the viscosity at both the low shear rate Newtonian region and the high shear rate power-law region.

For polymer melts, thermophysical properties such as density, heat capacity, and thermal conductivity do not change appreciably with temperature and pressure at normal extrusion conditions. Therefore, constant values are as- sumed in this study. For the polystyrene used, density is chosen as p = 1.1 x 103kg/m3, heat capacity is set at c, = 1.71 x 103J/kg-K, and thermal conductivity is chosen as k = 0.131 J/s-m-K (33).

Experimental Setup In order to vary the die wall temperature, a

die adapter was designed and installed in an Instron capillary rheometer. A schematic dia- gram of the setup is shown in Fig. 2. The cap- illary die is maintained at constant temperature by employing a band heater and a digital tem-

CAPILLARY DIE Fig. 2. Schematic diagram of experimental setup used in this study.

perature controller. The adapter is well insu- lated to minimize the temperature variation within 1°C. A thin gap between the adapter and the barrel serves as a thermal barrier to isolate the die wall temperature from the barrel tem- perature. Two flat entrance Instron capillary dies with LID ratios of 40 and 20, respectively, and a radius of 0.064 cm serve as flow chan- nels. Die wall temperature and melt tempera- ture (i.e., barrel temperature) were varied from 170°C to 230°C. Polymer melt flow rate was varied from 1.43 X to 5.72 X lop2 cm3/s by changing the crosshead speed of the Instron machine. Pressure drop across the die was mea- sured by an Instron load cell.

To simulate the profile extrusion, a two-hole die was built to study the effect of die wall temperature on the relative flow rate between the two circular channels. The length of the die is 3,066 cm and the radii of the two channels are 0.080 and 0.102 cm, respectively. The LID ratio of the smaller channel is 19.10, and the LID ratio of the larger channel is 15.05. The two channels are separated from each other by 0.25 cm. For some cases, the larger channel was capped and the two-hole die was used as a capillary die with LID = 19.10.

RESULTS AND DISCUSSION Pressure drops measured across the L / D = 40

die under isothermal conditions (2 10" to 230°C) are plotted in Fig. 3. The effect of heating (T, = 230"C, T, = 210°C) and cooling (T, = 210°C T, = 230°C) the die wall on the pressure drop of melt flow is shown in Fig. 4. Compared with the isothermal results, one can see that increas- ing the die wall temperature decreases the pres- sure drop and decreasing the die wall tempera- ture increases the pressure drop. If one com- pares the cooling and heating cases, it is clear

1080 POLYMER ENGINEERING AND SCIENCE, MID-AUGUST, 1987, Yo/. 27, NO. 14

Effect of Die Temperature on the Flow of Polymer Melts

O l . . . . ' " . . ' ' . " ' . . . . 1 0.0 i .5 3.0 4.5 8.0

Fig. 3. Pressure drop versus volumetricflow rate for LID = 40 die at isothermal conditions (0: 210°C. A: 220°C. +: 230 "C) .

FLOW RATE (cm**3/sec) X10**2

1

0.0 i .I 3.0 4.6 8.0 FLOW RATE (cmww3/secl X i 0 * * 2

Fig. 4 . Comparison of the effect of heating and cooling the die wall on pressure drops (a: T , = 210. T , = 230°C; A: T,,, = 230, T , = 210°C; --- isothermal results).

that die wall temperature has a greater effect on the pressure drop than melt temperature.

Pressure drops across the LID = 20 die versus flow rate at high temperatures (210" to 230°C) are shown in Fig. 5, while Fig. 6 is a plot of pressure drop across the LID = 19 die for tem- peratures ranging from 170" to 200°C. At high temperatures, the effect of die wall temperature on the pressure drop is the same for both long and short dies except that data points for the non-isothermal cases at high flow rates tend to move together. At low temperatures (i.e., Fig. 6 ) , a crossover of pressure drop curves is found for the short die when comparing the data points of the cooling and heating cases. This can be explained as die wall temperature at low flow rates has a greater effect on the pressure drop due to the long residence time of polymer melt in the flow channel. At high flow rates,

residence time of polymer melt is much shorter, consequently, the effect of die wall temperature is less significant and pressure drop is predom- inantly determined by the melt temperature. For long dies, such as LID = 40, the crossover of pressure drop curves may occur at much higher flow rates, which were not measured in this study.

For the two-hole die, when the die wall tem- perature was varied from the inlet polymer melt temperature, not only was the pressure drop altered, but the ratio of the two flow rates was also changed. The ratios of the flow rates through the small channel to those through the large channel (&I&) are presented in Table 1 at both low and high temperatures. Results in- dicate that varying the die wall temperature changes the flow rate ratio between the two channels. In the case of heating the die wall

0.0 1.5 3.0 4.5 6.0

FLOW RATE (cma/sec)X1 0.

Fig. 5. Pressure drop versus volumetricflow rate for LID = 20 die at various temperatures (+: T, = 21 0. T, = 21 0°C; *: T , = 210, T, = 230°C; 0: T , = 230, T, = 210°C; X: T,,, = 230, T , = 230°C).

0 s * b

0 0 a 0

a

n W 0 K n 3 In

K

r X n

wl-

0 E

E n

0.0 1.5 3.0 4.5 6.0

FLOW RATE (cma//sec)X 10'

Fig. 6. Pressure drop versus volumetricflow rate for L/D = 19 die at temperatures ranging from 170°C to 200°C (+: T , = 190, T, = 170°C; *: T, = 170, T, = 190°C; 0: T , = 200, T , = 180°C; X: T, = 180, T , = 200°C).

POLYMER ENGINEERING AND SCIENCE, MID-AUGUST, 1987, Yo/. 27, No. 14 1081

B. Yang and L. J a m e s Lee

Table 1. Flow rate ratio (Qs/QL) for two-channel die at both low and high temperatures.

A. Qs/QL at High Temperatures

Die Melt Temperature

("C) (cm3/s) 210°C 230°C Temp. Q t o ~

0.0572 0.280 0.260 21 0 0.0286 0.297 0.269

0.0143 0.305 0.290

0.0572 0.31 1 0.274 220 0.0286 0.322 0.297

0.0143 0.338 0.300

0.0572 0.332 0.307 230 0.0286 0.348 0.321

0.0143 0.355 0.339

8. Qs/QL at Low TemDeratures

Die Melt Temperature

("C) (cm3/s) 170°C 180°C 190°C

0.0572 0.255 0.209 0.173 170 0.0286 0.249 0.215 0.188

0.0143 0.250 0.218 0.210

0.0572 0.258 0.231 180 0.0286 0.261 0.237

0.01 43 0.271 0.257

Temp. Qt-i

0.0572 0.325 0.290 0.252 190 0.0286 0.321 0.299 0.269

0.0143 0.319 0.294 0.282

(i.e., T, > Tm), the flow rate ratio increases, which implies that more material flows through the smaller channel. On the other hand, in the case of cooling the die wall (i.e., T, < T,), the flow rate ratio decreases, which implies that polymer melt prefers to flow through the larger channel. This is apparently due to the fact that melt viscosity is highly temperature dependent. When die wall temperature is changed, the vis- cosity of polymer melt near the wall will be affected. Such an influence is greater for the smaller channel because of the smaller cross sectional area. Therefore, when the die wall temperature is increased, the flow resistance is reduced more in the smaller channel than in the larger channel and, consequently, more ma- terial will flow through the smaller channel. Similar reasoning holds for the case of cooling the die wall.

The data listed in Table 1 show that melt temperature and total flow rate may also affect the flow rate ratio. Increasing total flow rate or decreasing melt temperature reduces the flow rate ratio. The influence is quite significant at high temperatures but not very noticeable at low temperatures. Shear thinning and temper- ature dependency of melt viscosity are the rea- sons why the flow rates may vary when one of the three variables-die wall temperature, melt temperature, or total flow rate-changes. Of the three, varying the die wall temperature has the greatest influence on the flow rate ratio, especially at low temperatures.

NUMERICAL SIMULATION Fully Developed Flow

In order to predict temperature, velocity and pressure profiles of polymer melts flowing through confined channels under non-isother- ma1 conditions, equations of continuity, mo- mentum, and energy need to be solved. Because of the coupling of these nonlinear differential equations, solutions by analytical approaches are impossible. The typical approach is to use numerical techniques to solve for these profiles.

Let us consider a homogeneous and isotropic polymer melt with a uniform temperature, T,, flowing into a cylinder of diameter D and length L. The temperature of the cylinder wall is kept at a constant value, T,, and the volumetric flow rate of polymer melt is fixed at 9. Major as- sumptions for the flow are as follows:

1.

2.

3. 4. 5.

6. 7.

8. 9.

10.

11.

12.

The

The flow is laminar, steady state, and axially symmetrical. Thermal conductivity, k, and heat capac- ity, c,. are independent of temperature and pressure. The fluid is incompressible. There is no slip at die wall. Viscosity can be expressed as a Carreau viscosity equation (i.e., E q 1 ) Gravitational forces are negligible. Inertial effects in the momentum equa- tion are negligible. Axial conduction can be neglected. The radial (i.e., r) component of velocity is important only in the convection of energy and the continuity equation. Pressure is uniform in the r and 0 direc- tions; i.e., only the pressure gradient in the flow (i.e., z ) direction needs to be considered. Normal stresses due to acceleration as well as elasticity are negligible. Fully developed non-Newtonian isother- mal flow exists at the entrance of the cylinder. first nine assumptions are typical in the

analysis of many polymer process& Assump- tions 10 to 12 lead us to neglect the entrance and exit effects which, however, are important for polymer flow in short channels. Neglecting these effects may cause errors in the calculation of pressure drop across the die. The end effect in channel flows will be discussed in the next section.

With the above mentioned assumptions, the equations of change in cylindrical coordinates can be reduced to:

ap I a(rT,,) az r ar

Eq. of Motion - + - ~ = 0

Eq. of Energy pc, u, - + u, - 1 zf ar

1082 POLYMER ENGINEERING AND SCIENCE, MID-AUGUST, 1987, VOI. 27, NO. 14

Effect of Die Temperature O R t h e Flow of Polymer Melts

.* 100%

0 21-

x 230% t

i

t

*t

3*

r x o

x o

L

d(mJ duz ar az Eq. of Continuity ~ + r - = 0 (4)

dv, T 2 where T, = - 4 - and @ = -E which is the

dr D viscous heating term. The applicable boundary conditions are:

r = 0 , O S z S L d T - = o d r

7, = 0 r = 0 , O S z S L

T = T, r = R , O l z S L

u, = 0 r = R , O S z S L

u, = 0 r = R , O S z S L

P = o z = L , O S r S R

T =T, z = O , O S r l R

u, = 0 z = O , O S r S R

u, = uo(r) z = 0, 0 S r S R

where uo(r) is assumed as a fully developed flow at the die entrance with temperature T,.

An implicit finite difference method was used to solve the equations, where three point im- plicit central difference approximation was used in the r direction, while two point back- ward difference approximation was used in the z direction. One hundred grid points were used in both the r and z directions. Because the final form of finite difference equations are nonlin- ear, a Picard type iteration scheme was used.

Since temperature gradient near the die wall and the inlet region is higher than that in other locations, a finer grid is used in these regions to facilitate convergence of the numerical solu- tion. The independent variables r and z are transformed exponentially into two new varia- bles w and s, and expressed as

w = exp(ar) (51

(6)

and s = exp[P(l - z ) ]

For the simulation of flow through the two- hole die, pressure drop was determined by iter- ating the calculation between two channels un- til both channels had the same pressure drop. In order to accelerate the iteration, a relaxation factor was used. Detailed numerical scheme is discussed elsewhere (34).

End Effect The prediction of end effect has been the

subject of many studies (35-38), but calcula- tions based on theoretical models still cannot provide satisfactory results due to the limitation of applying viscoelastic constitutive equations in advanced numerical methods such as the finite element method. In order to predict ac-

curately the pressure drop across the die, end pressure drops at different temperatures and shear rates need to be known. Bagley’s method (39) was carried out to obtain end pressure drops at isothermal conditions.

Figure 7 plots the end pressure drop against wall shear stress at isothermal conditions for three capillary dies with radius equal to 0.064 cm. End pressure drops fall roughly into a single curve when plotted against wall shear stress and are independent of melt temperature. Data points at low shear rates and high temperatures tend to scatter somewhat. This is due to the limitation of the load cell used in the Instron rheometer. These data points, with magnitude less than 1.124N (5 lbf), have reached the ac- curacy limit of the 224.8N (1000 lbf) load cell used.

Since the mechanism of entrance pressure drop is still not well-understood, a power func- tion is used here to describe the relationship between end pressure drop and wall shear stress. I t can be expressed as

M e n d = 0.763 Tw1.63 (7) where units for both wall shear stress and end pressure drop are in kPa. It is interesting to notice that the exponential dependence, 1.63, happens to agree well with that proposed by Oda e t al. (40) for the correlation of first normal stress difference and wall shear stress for polystyrenes.

Equat ion 7 is based on isothermal results only, because Bagley’s method cannot be used for the determination of end pressure drops in non-isothermal conditions. In this study, we assume that Eq 7 is applicable for both isother- mal and non-isothermal conditions.

Shaw (41) discussed the applicability of using end pressure drops obtained from dies at one radius to dies with different radii. He claimed that end effect can be evaluated only for a set

0

$ I

170%

Fig. 7 . End pressure drops versus wall shear stress at isothermal conditions forflat entrance dies with radius equal to 0.064 cm.

POLYMER ENGINEERING AND SCIENCE, MID-AUGUST, 1987, Vol. 27, NO. 14 1083

B. Yang and L. James Lee

N

b

of capillaries with equal radii since, in the Weissenberg-Rabinowitsch-Mooney relation l . . . I ' . ' I . . .

. + Tm-210. Tw-230.C

. x Tm-230. Tw-210%

(8) 3 4Q rw d(4Q7rR3) 4 r R 3 4 drw '

(+)w = - - + -

N

b 7

the shear rate at die wall is not the same as the pseudo-shear rate, ~ Q / T R , ~ used in the calcu- lation of the end effect. Therefore, the end ef- fect would be different for capillaries with dif- ferent radii.

However, if the viscosity of polymer melts can be well described by the power-law model, E q 8 can be reduced to

, * . ' I . ' .

.t 210%

- * 220%

. O 230%

1 + 3 n 4Q (+)w = ~ - 4n (7rR3)

where n is the power-law index. If n is a con- stant value, it is reasonable to assume that the shear rate at die wall is proportional to the pseudo-shear rate and the end pressure drop measured by dies with one radius can be used for dies with different radii. In this study, we assume that Eq 7 is applicable for all dies used, which may cause some errors at low flow rates.

Numerical Results The simulated pressure drop versus flow rate

at isothermal conditions for the LID = 40 die is plotted in Fig. 8, where the simulated results (solid lines) match very well with experimental data. Dashed lines in the figure are the predic- tion without considering end pressure drops. At high shear rates, the differences (i.e., end pres- sure drops) are about 10% of the total pressure drop. Similar plots for non-isothermal condi- tions (both heating and cooling cases] are shown in Fig. 9. Again, the simulation results (solid lines) match well with experimental data and the end pressure drops are about 10% of the total pressure drop.

Figure 10 shows the non-isothermal results

t - S'MUuTloN 1

0.0 1 .5 3.0 4.5 6.0

FLOW RATE (cma/sec)X1 Oa

Fig. 9. Predicted pressure drop uersusjlow rate at non- isothermal conditions with LID = 40 die [- simulation. _ _ _ sirnulation without considering end effect].

, . . . I . . . , . . .

. + Tm-210. Tw-230%

. * Tm-230. Tw-210%

.-SIMULATION

0.0 1.5 3.0 4.5 6.0

FLOW RATE (crna/sec)X1 0'

Fig. 10. Predicted pressure drop uersusjlow rate at non- isothermal conditions with LID = 20 die [- simulation, _ - _ simulation without considering end effect).

for the LID = 20 die. Again, the predicted pres- sure drops match well with experimental data. The magnitude of end pressure drop does not change with different LID ratio. Therefore, the end pressure drop is about 20% of the total pressure drop for the LID = 20 die. From Figs . 9 and 10, it seems that the end effect at non- isothermal conditions can be reasonably esti- mated from the isothermal correlations (i.e., Eq. 7).

Figure 1 1 plots simulated isotherms along the axial direction. If we define the heat penetration layer as the thickness of the layer where dimen- sionless temperature, T = (T - Tm)/(Tw - Tm), is higher than a certain value, then the curves shown in Fig. 11 are indicative of the change of the thermal boundary layer in the flow direc- tion. For small value of T and long dies (i.e., L/D = 40), such a layer has merged in the center before the flow reaches die exit. Figure 12 plots the velocity profile near die exit for both iso-

1084 POLYMER ENGINEERING AND SCIENCE, MID-AUGUST, 1987, Vol. 27, No. 14

Effect of Die Temperature on the Flow of Polymer Melts

0 9 .-

5 x 6 2

F

K O

0

2 - 0.00 0.25 0.50 0.75 1 .oo

FLOW DIRECTION

Fig. 1 I. Simulated temperature profiles along the axial direction at non-isothermal conditions.

I -

PSEUDO SU€%R RATE - 742 am<'

4 c

I > 0

2 -

q 0

9 0

0.00 0.25 0.50 0.75 1 .oo RADIAL DIRECTION

Fig. 12. Simulated velocity profiles near die exit at iso- thermal and non-isothermal conditions.

thermal and non-isothermal conditions. As ex- pected, heating of the die wall flattens the ve- locity profile. On the other hand, cooling of the die wall makes the velocity profile more para- bolic. Therefore, the velocity gradient at die wall is higher for the heating cases and lower for the cooling cases.

Model prediction of flow rate ratios for the two-hole die is plotted in Fig. 13. Solid lines are the simulation result, while symbols are exper- imental data. The simulation results match rea- sonably well with experimental data. If end pressure drops are not considered in the simu- lation, which are expressed as dashed lines in the figure, the predicted flow rate ratios differ significantly from experimental data but the trend is still correct.

Viscous heating is a very important phenom- enon for polymer melts flowing in confined

channels, which can decrease the pressure drop and affect the quality of extrudate. Figure 14 shows the temperature distribution of polysty- rene melt in the radial direction at die exit (LID = 40) under different flow rates [i.e., pseudo- shear rates), where die wall temperature and melt temperature at the entrance are both 2 10°C. The results indicate that viscous heating at shear rates lower than 142 s-', which is typical for extrusion processes, is not very prominent. However, viscous heating effects cannot be neglected at high shear rates (e.g., 284 s-' or higher) such as in the case of injection molding.

Figure 15 plots temperature profiles with and without considering viscous heating in the ra- dial direction at two axial positions for both the heating and cooling cases at i. = 142 s-'. The temperature scale used in the plot is normal- ized. Results indicate that viscous heating in- creases the heating effect in the heating case,

+ Tm-230, Tw-23OoC

~t Tm-230. Tw-210

0 Tm-210. Tw-230

X Tm-210. Tw-210

0.00 0.0 1 0.03 0.04 0.06

TOTAL FLOW RATE (crna/soc)

Fig. 13. Model prediction offlow rate ratios for the two- hole die (- simulation. - - - simulation without consid- ering end eflect).

: N 1

0.00 0.20 0.50 0.75 1 .oo RADIAL DIRECTION

Fig. 14. Simulated temperature distribution near die exit in the radial direction at dqferent shear rates (LID = 40, T,,, = T, = 210 "C).

POLYMER ENGlNEERl" AND SCIENCE, MID-AUGUST, 1987, Vol. 27, No. 14 1085

B. Yang and L. James Lee

while it decreases the cooling effect in the cool- ing case.

The effect of viscous heating on velocity pro- files, however, is much less than that on tem- perature profiles. Figure 16 plots the velocity profiles near the die exit at i. = 142 s-' with and without considering viscous heating. The figure indicates little difference in velocity profiles.

CONCLUSION In this work, we analyzed the non-isothermal

flow of polymer melt in circular channels. An implicit finite difference method was used to describe the effect of die wall temperature on the flow of polymer melts. The model success-

-.-WITHOUT VISCOUS HEATING - Tm-25O0C,Tw-21 ooc

n E I-

- -Trn-21O0C.Tw-23O0C

0.00 0.2s 0.so 0.75 1 .oo RADIAL DI R ECTl ON

Fig. 15. Simulated temperature profiles in the radial di- rection at two axial positions for heating and cooling cases [i. = 142 s-').

I> \ >

9 r

9 r-

'? 0

0 0

Tm=Tw=21 ooc

Trn=21 0,Tw=230°C

WITH VISCOUS HEATING

WITHOUT VISCOUS HEATING

0.00 0.25 0.50 0.75 1 .oo RADIAL DI R ECTl O N

Fig. 16. Simulated velocity profiles near die exit with and without considering viscous heating [+ = 142 s-l).

fully predicts die pressure changes and flow rates under different wall temperatures for var- ious capillary dies. Combined with the experi- mental results, we conclude that:

1. Increasing the die wall temperature de- creases the die pressure drop when the flow rate of the polymer melt is kept constant. Die temperature has a greater effect on the pressure drop than that of melt temperature.

2. For the two-hole die, because of the tem- perature-dependent viscosity, increasing the die wall temperature tends to shift more mate- rial flowing through the small channel, while decreasing the die wall temperature tends to shift more material flowing through the larger channel.

A, B

k L n N1 P

Q QL

QS r R

w e n d

S

NOMENCLATURE = constants in Eq 1. = heat capacity. = diameter of die. = thermal conductivity. = length of die. = power-law index. = first normal stress difference. = pressure. = entrance pressure drop. = flow rate. = flow rate through the large channel. = flow rate through the small channel. = radial direction. = radius of die. = variable z in exponential form, i.e., E q

= melt temperature. = die wall temperature. = dimensionless temperature = (T -

= velocity. = velocity profile at die entrance. = variable r in exponential form, i.e.,

= flow direction. = constant in E q 5. = constant in E q 6. = shear rate. = wall shear rate. = shear viscosity. = constant in E q 1. = zero shear rate viscosity at tempera-

= density. = shear stress = -Vdu,/dr. = wall shear stress. = viscous heating term.

REFERENCES

6.

Trn)/(Tw - Trn).

Eq 5.

ture T.

1. R. J. Brown, H. T. Kim, and J. W. Summers, SPE ANTEC Tech. Paper, 25, 130 (1979).

2. J. I. Fulton and H. H. Hopfe, U S . Patent, No. 4,332,543 (1982).

3. L. J. Lee, J. F. Stevenson, and R. M. Griffith, U. S. Patent, No. 4,425,289 (1984).

4. J. F. Stevenson. L. J. Lee. and R. M. Griffith, Polyrn. Eng. Sci.. 26. 233 (1986).

1086 POLYMER ENGINEERING AND SCIENCE, MID-AUGUST, 1987, Yo/. 27, NO. 14

Effect of Die Temperature on t h e Flow of Polymer Melts

5. H. C. Brinkman, Appl . Sc i . Res . . A2, 120 (1951). 6. R. B. Bird, SPE J., 11, 35 (1955). 7. H. L. Toor, AIChE J.. 4, 319 (1958). 8. C. Tien, Can. J. Chern. Eng.. 39. 45 (1961). 9. N. Mitsuishi and 0. Miyatake, Intern. Chern. Eng., 9.

352 (1 969). 10. J . Vlachopoulos and C. K. J . Keung, AIChE J., 18, 1272

(1972). 11. J . A. Prins, J . Mulder, and J. Schenk, Appl . Sci. Res . ,

A2, 431 (1951). 12. J . C. Lyche and R. B. Bird. Chern. Eng. Sci. , 6, 35

(1956). 13. C. Tien, C a n . J. Chern. Eng., 40, 130 (1962). 14. W. H. Suckow, P. Hrycak, and R. G. Griskey, Polyrn.

Eng. Sci. , 11, 401 (1971). 15. S. Matsuhisa and R. B. Bird, AIChE J.. 11, 588 (1965). 16. R. Mahalingam, S. F. Chan, and J . M. Coulson, Chern.

Eng. J., 9. 161, (1975). 17. R. Mahalingam, L. 0. Tilton, and J. M. Coulson. Chern.

Eng. Sci. , 30, 921 (1975). 18. R. E. Gee and J . B. Lyon. Ind. Eng. Chern., 49. 956

(1957). 19. E. B. Christiansen, G. E. Jensen, and F. S. Tao, AIChE

J.. 12, 196 (1966). 20. J. E. Gerrard, F. E. Steidler, and J . K. Appeldoorn, I &

EC Fundarn., 4.332 (1965). 21. J. E. Gerrard. F. E. Steidler. and J . K. Appeldoorn, I &

ECFundarn. , 5.260 (1966). 22. J. Gavis and R. L. Laurence, I & EC Fundarn., 7, 525

(1968). 23. H. T. Kim and E. A. Collins, Polyrn. Eng. Sci. , 11, 83

(1 97 1).

24. H. W. Cox and C. W. Macosko, AIChE J., 20,785 (1 974). 25. N. Galili, R. Takserman-Krozer, and Z . Rigbi, Rheol.

Ac ta . , 14, 550 (1975). 26. E. E. Agur and J. Vlachopoulos, J . Appl . Polyrn. Sci. ,

26, 765 (1981). 27. R. A. Morrette and C. G. Gogos, Polyrn. Eng. Sci. , 8,

272 (1968). 28. J . F. T. Pittman, and S . Nakazawa, Polyrn. Eng. Rev.,

4. 143 (1984). 29. J. E. Gerrard, J . K. Appeldoorn, and W. Philippoff,

Nature , J u n e 16, 1067 (1962). 30. R. G. Griskey and I . A. Wiehe, AIChE J., 12,308 (1966). 31. C. E. Bassett and J . R. Welty, AIChE J. , 21,699 (1975). 32. P. J. Carreau, Trans . SOC. Rheol. , 16, 99 (1972). 33. D. W. Van Krevelen and P. J. Hoftyzer, "Properties of

Polymers , The i r Es t imat ion a n d Correlation w i t h Chemical Structure," Elsevier Scientific Publishing Co., N.Y. (1976).

34. B. Yang, Ph.D. Dissertation, The Ohio State University (1986).

35. D. V. Boge, R. Gupta, and R. I. Tanner, J . Non-Newt. Fluid Mech., 4, 239 (1978).

36. C. D. Han, AIChE J., 16, 1087 (1970). 37. C. D. Han, AIChE J., 17, 1480 (1971). 38. J . L. Duda and J . S. Vrentas, Trans . SOC. Rheol. . 17,

89 (1973). 39. E. B. Bagley, J. Appl . Phys . , 28, 624 (1 957). 40. K. Oda, J. L. White, and E. S . Clark, Polyrn. Eng. Sc i . ,

18.25 (1978). 41. M. T. Shaw, SPE ANTEC Tech . Paper , 32,707 (1986).

POLYMER ENGlNEERlNG AND SCIENCE, MID-AUGUST, 1987, VOI. 27, NO. 14 1087